Due Mar 23, 3:59 AM EDT
A graph has 100 vertices and only one edge. How many connected components does it have?
Indeed, the only edge connects two vertices, which form one connected component. The remaining 98 vertices are isolated, thus each of them forms its own component.
Compute the circuit rank of the following graph.

One can break three edges and get a tree. Breaking two edges is not sufficient.
A graph has 100 vertices and only 50 edges. What is the smallest possible number of connected components in such a graph?
The number of connected components is always greater or equal than the number of vertices minus the number of edges. Thus, it is at least 50. A graph with 100 vertices, 50 edges and 50 connected components can be constructed, for example, in the following way: take 50 pairs of vertices and connect the vertices in each pair by an edge.
Compute the circuit rank of a graph which has 7 vertices and 16 edges.
The graph should be connected: otherwise, the number of edges could not be greater than 15 (this is the best case for a disconnected graph on 7 vertices, with one component of 6 vertices and one isolated vertex). Thus, the circuit rank is computed as 16 - 7 + 1 = 10.